State space representation

transfer function to a state space representation

Given

G(s) = \frac{\sum_{i=1}^{n-1}b_is^i + b_{0}}{s^n + \sum_{i=1}^{n-1}a_is^{i} + a_{0}} = \frac{Y(s)}{U(s)}

We get controller canonical state space form:

\begin{aligned} \dot{x}(t) &= \begin{bmatrix} 0 & 1 & 0 & \cdots & 0 & 0 \\\ 0 & 0 & 1 & \cdots & 0 & 0 \\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\\ 0 & 0 & 0 & \cdots & 1 & 0 \\\ 0 & 0 & 0 & \cdots & 0 & 1 \\\ -a_0 & -a_1 & -a_2 & \cdots & -a_{n-2} & -a_{n-1} \end{bmatrix} x(t) + \begin{bmatrix} 0 \\\ 0 \\\ \vdots \\\ 0 \\\ 0 \\\ 1 \end{bmatrix} u(t) \\\ y(t) &= \begin{bmatrix} b_0 & b_1 & \cdots & b_{n-2} & b_{n-1} \end{bmatrix} x(t). \end{aligned}

We get observer canonical state space form:

\begin{aligned} \dot{x}(t) &= \begin{bmatrix} -a_{n-1} & 1 & 0 & \cdots & 0 & 0 \\ -a_{n-2} & 0 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ -a_2 & 0 & 0 & \cdots & 1 & 0 \\ -a_1 & 0 & 0 & \cdots & 0 & 1 \\ -a_0 & 0 & 0 & \cdots & 0 & 0 \end{bmatrix} x(t) + \begin{bmatrix} b_{n-1} \\ b_{n-2} \\ \vdots \\ b_2 \\ b_1 \\ b_0 \end{bmatrix} u(t) \\ y(t) &= \begin{bmatrix} 1 & 0 & \cdots & 0 & 0 \end{bmatrix} x(t). \end{aligned}